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Let $a$ and $b$ be the roots of the quadratic equation $2x^2 - 7x + 2 = -x^2 + 4x + 9.$ Find $\frac{1}{a+7}+\frac{1}{b+7}.$

 Jan 14, 2024
 #1
avatar+289 
+1

Simplify:

 

3x^2 - 11x - 7 = 0

 

Finding the roots by plugging in the values a = 3, b = -11, and c = -7 into the quadratic equation \(x = {-b \pm \sqrt{b^2-4ac} \over 2a}\), we get:

 

x = \(\frac { \sqrt{205} + 11}{6}\), x = \(\frac { -\sqrt{205} + 11}{6}\)

 

We now have that a = \(\frac { \sqrt{205} + 11}{6}\) and b = \(\frac { -\sqrt{205} + 11}{6}\).

 

Plugging in these values for the expression \(\frac{1}{a+7}+\frac{1}{b+7}\), we get:

 

\(\frac{1}{\frac { \sqrt{205} + 11}{6}+7}+\frac{1}{\frac { -\sqrt{205} + 11}{6}+7}\)

 

Simplifying this expression, we get:

 

\(\frac{1}{\frac { \sqrt{205} + 53}{6}}+\frac{1}{\frac { -\sqrt{205} + 53}{6}}\)

 

Removing the 6 and multiplying 1 by the reciprocal of \({\frac { \sqrt{205} + 53}{6}}\), we get:

 

\(\frac{6}{ \sqrt{205} + 53}+\frac{6}{ -\sqrt{205} + 53}\)

 

We can plug each of these terms in the calculator to get

 

\(0.089 + 0.155\)

 

Adding these two values, we get:

 

\(0.244\)

 

Answer: 0.244

 Jan 14, 2024
 #2
avatar+128732 
+1

\($\frac{1}{a+7}+\frac{1}{b+7}.$ \)

 

Simplify  as

 

3x^2 -11x -7  =  0

 

We have the form mx^2 + nx + c  = 0

 

m =3 , n = -11   c = -7

 

Product of  roots   = ab = c/m   =  -7/3

 

Sum of  roots =  

 

a + b = -n/m  =   11/3          

 

1/ (a + 7)  +  1 /(b + 7)   =

 

[ a + 7 + b + 7 ] / [ (a + 7) (b + 7) ]  =

 

[ a + b + 14 ] / [ ab + 7 (a + b) + 49 ]  = 

 

[ 11/3 + 14 ]  / [ -7/3 + 7(11/3) + 49 ]  =

 

[ 53/3 ] / [ 217 /3 ]  =

 

53 / 217

 

cool cool cool

 Jan 14, 2024

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